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Static Light Scattering technologies for GPC -SEC explained

Contents

Introduction

Introduction• What is molecular weight?• What is molecular size?• What is light scattering?

Static Light Scattering Theory• How is molecular weight related to light scattering?• What is static light scattering?• Static light scattering in GPC/SEC

Different light scattering technologies• An introduction to angular dependence• How does angular dependence affect our calculations?• How do we get the molecular size, Rg, from the data?

Light scattering instruments• Right-angle light scattering (RALS)• Low-angle light scattering (LALS)• Multi-angle light scattering (MALS)• Summary of the benefits of MALS, LALS, and RALS systems

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2 Static Light Scattering technologies for GPC - SEC explained

Calibration• Are static light scattering measurements 'absolute'?• Principle of instrument calibration• Molecular weight standard-based calibration• Scattering standard-based calibration

Useful references

SummaryStatic light scattering is a technique to measure the molecular weight using therelationship between the intensity of light scattered by a molecule and its molecularweight and size. These relationships are described by Rayleigh theory which statesthat the molecular weight of a molecule is proportional to the Rayleigh ratio of scatteredlight i.e. the ratio of scattered light intensity to incident light intensity.

All static light scattering instruments detect the amount of light scattered by a sampleto measure its molecular weight; however, as molecules grow in size, a second factorcalled angular dependence becomes significant. Angular dependence affects theintensity of scattered light and hence the calculated molecular weight. It must thereforebe accounted for.

• A RALS detector collects scattered light at 90°. It can measure molecular weightwith high sensitivity for samples that scatter light isotropically i.e. equally in alldirections; however it cannot measure molecular weight for anisotropic scatterers(those affected by angular dependence).

• A LALS detector collects scattered light at 7°. It can measure molecular weight forall molecules but has a lower signal-to-noise.

• By combining RALS and LALS detectors into a hybrid system the molecularweight of all samples can be measured while maximising signal-to-noise whererequired. It therefore offers the strengths of both LALS and RALS with none of theirweaknesses.

• A MALS detector collects scattered light at many angles. This data is used to modelthe angular dependence to account for it in the molecular weight calculation. Itcan measure molecular weight for both isotropic and anisotropic scatterers and foranisotropic scatterers can also measure the radius of gyration.

• All static light scattered instruments must be calibrated before use. Calibration caneither be performed using a molecular weight standard or a scattering standardand each has advantages and disadvantages that are described in the calibrationsection.

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Glossary of AcronymsGPC Gel permeation chromatography

SEC Size exclusion chromatography

LALS Low angle light scattering

RALS Right angle light scattering (90° light scattering)

MALS Multi angle light scattering

SLS Static light scattering

DLS Dynamic light scattering

Rg Radius of gyration

RH Hydrodynamic radius

MW Molecular weight (Molar Mass)

Mn Number-average molecular weight

Mw Weight-average molecular weight

Mz Z-average molecular weight

A2 Second virial coefficient (B22)

IntroductionLight scattering can be a confusing topic. There are a number of different techniquesthat come under the general heading of light scattering and a range of parameters thatcan be measured. The lines between these can become blurred, as some systems usemore than one technique in a single instrument, so it is very easy to lose sight of themost important factors when considering which light scattering technology is right foryour application.

Static light scattering, also called classical light scattering, is a technique usedto measure molecular weight and molecular radius of gyration. Within static lightscattering, there are a number of different technologies with acronyms such as SLS,MALS, LALS and RALS and others. Each of these is subtly different and each hasadvantages and disadvantages.

This white paper describes, in some detail, the principles and theories behind staticlight scattering. The theory, technique and the different technologies are describedand compared. By reading this white paper, you should be able to understand thebackground to these measurements and make an informed choice on the mostappropriate solution for your application.

Each section provides a great deal of detail, but begins with a few key points tosummarise the information contained so you can decide whether you need to read thedetail in that section.

What is molecular weight?

Key Points• Molecular weight is the molar mass of the molecule under study.• There are a number of moments that can be calculated from a molecular weight

distribution including Mn, Mw and Mz.• Polydispersity describes the broadness of the molecular weight distribution.

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DetailsMolecular weight is a property that describes the mass of an individual molecule.Scientifically, it is the mass of a material required to make 1 mole of the sample. Forexample, carbon has a molecular weight 12 g/mol. The units of molecular weightare grams per mole (g/mol). However these are often referred to as Daltons (Da), orsometimes more conveniently kilo Daltons (kDa)

For a pure protein sample, it is expected that it would have a fixed molecular weightand that all of the protein molecules within the sample should have the same molecularweight. For a natural or synthetic polymer sample, the molecular weight of themolecules within a particular sample will have a distribution that could have manyforms. The term that describes the overall width of the distribution is 'polydispersity'.A sample with a very narrow range of molecular weights is said to have a lowpolydispersity or to be very monodisperse. A sample with a wider range of molecularweights is said to have a high polydispersity.

The majority of samples contain a range of molecular weights, particularly naturaland synthetic polymers. These distributions of molecular weights can be describedin a number of ways, and one common way is to use the molecular weight momentsMn, Mw and Mz, which are the number-averaged, weight-averaged and z-averagedmolecular weights respectively. These are defined by equations shown below, where ciand Mi are the concentration and molecular weight at each data slice:

Intuitively, the number-averaged molecular weight is the total mass of material dividedby the total number of molecules and is therefore a numerical average. It is weightedtowards the lower mass molecules which will be more numerous. Properties such asvapour pressure lowering, freezing point depression and osmotic pressure depend onthe number of molecules and not on their size. The weight average molecular weightis calculated by multiplying by the molecules mass, so the average is biased towardsthe larger molecules in the distribution. Properties like diffusion, sedimentation, lightscattering depend both on the size and the mass of the molecules. The z-averagemolecular weight includes a further multiplication of the molecules' molecular weight.It is therefore heavily weighted to the largest molecules in the sample and can bedetermined directly by the ultracentrifugation technique.

Using these three values, it is possible to get an idea of the entire molecularweight distribution. Clearly, the closer these values are to each other, the lowerthe polydispersity and the further apart they are, the higher the polydispersity.Polydispersity is defined according to the equation

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Since Mw is always equal to or larger than Mn, the lowest possible value forpolydispersity (Pd) is 1 but there is no theoretical upper limit. Natural and syntheticpolymers have a wide range of polydispersities but since proteins have fixed molecularweights, we can expect the polydispersity of a protein to be very low (close to 1) forwell-behaved proteins.

What is molecular size?

Key Points• Molecular size is the physical size of the molecule.• When discussing size from static light scattering alone, we are discussing Rg, the

radius of gyration.• The units of size are in nanometers.

DetailsMolecular size is the physical size of a molecule. Typically, a single value is usedto describe the size. This value is the radius of a sphere of an equivalent size to themolecule being measured. The two most commonly used values of molecular size areRH (hydrodynamic radius) and Rg (radius of gyration). The hydrodynamic radius is theradius of an equivalent sphere that diffuses at the same speed as the molecule understudy. It is calculated either from dynamic light scattering or from intrinsic viscosity andso will not be considered further here. (For more information on hydrodynamic radiussee the technical notes, "Dynamic Light Scattering: An Introduction in 30 Minutes" and"Protein and polymer molecular size by GPC/SEC", on the Malvern website).

The radius of gyration is the root-mean-square of the radii from the centre of mass tothe different mass cores within the molecule. It is sometimes called Rrms (root-mean-square). The units for a molecular size are nanometers (nm). Rg is calculated usingstatic light scattering so will be considered in more detail later in the document.

What is light scattering?

Key Point• When light hits a molecule or particle, some of that light is absorbed and re-emitted

in all directions.

DetailsWhen a photon collides with a molecule, some of the energy from the photon isused to initiate what is called an oscillating dipole within the molecule. This energyis subsequently re-emitted by the molecule in all directions as light. We see thisphenomenon every day in white clouds, sunsets or in dust passing through a beam ofsunlight or light from a projector. The principles behind light scattering can be used tomeasure a number of properties related to the molecule.

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Static Light Scattering Theory

How is molecular weight related to light scattering?

Key Point• Rayleigh theory describes the relationship between the intensity of the light

scattered by a sample and its size and molecular weight.

DetailsThe properties of the light scattered by molecules and particles vary depending on theobject doing the scattering as defined in the Rayleigh equation.

Where:

• C = the sample concentration,• θ is the measurement angle,• Rθ = the Rayleigh ratio (the ratio of scattered light intensity to incident light

intensity) at the measurement angle θ• Mw = Weight average molecular weight,• A2 = the second virial coefficient.• K and Pθ are more complex terms defined as:

Where:

• λ0 is the laser wavelength in a vacuum,• NA is Avogadro's number,• n0 is the refractive index of the solvent,• dn/dc is the refractive index increment of the sample.

Where:

• Rg is the molecule's radius of gyration

In simpler terms, the Rayleigh equation tells us that the intensity of scattered light at agiven angle is dependent on a number of factors including both molecular weight andmolecular size of the sample under study. As can be seen in the Rayleigh equation,molecules with higher molecular weight and larger sizes will scatter more light. Whilethe increase in the intensity of light scattered is linear with molecular weight, it is non-linear with respect to size.

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So, if we know all of the other factors in the Rayleigh equation, we can measure theintensity of the scattered light (related to Rθ) and calculate the sample's molecularweight.

What is static light scattering?

Key Point• Static light scattering is the technique we use to measure molecular weight using

light scattering.

DetailsStatic light scattering (SLS) uses an optical arrangement such that the signal detectedis 'static' or stable. By measuring the intensity of light scattered by a sample insituations where the other constants are known, the molecular weight of the samplecan be calculated.

The term static light scattering is used to differentiate the technique from dynamic lightscattering which is a separate, though not unrelated, technique for measuring particlesize. For more information on dynamic light scattering, see the Malvern technical note"Dynamic Light Scattering: An Introduction in 30 Minutes" or the 'inform' white paper "Abasic guide to particle characterization" on the Malvern website.

Static light scattering in GPC/SEC

Key Points• SLS measurements are most commonly performed as part of a GPC/SEC system

but can be made in cuvettes as well.• To perform SLS measurements, a concentration detector is also required which is

typically RI but can also be UV.

DetailsStatic light scattering measurements can be made in a cuvette or in a gel-permeationchromatography (GPC)/size-exclusion chromatography system (SEC). Using achromatography system removes a number of problems to do with preparing andpurifying the sample and so is the most common implementation of SLS. Additionally,GPC/SEC allows us to easily combine the light scattering data with data from aconcentration detector to measure the sample's concentration at the same time andthus use both sets of information in our calculations. The most common concentrationdetector is a refractive index detector (RI) but an ultraviolet (UV) absorbance detectorcan also be used.

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Figure 1: Schematic of a GPC/SEC system. The degasser, pump, injection loop and columns are standard withthe optional autosampler. After separation, the molecules are measured by one or more detectors in sequence,in this case, a light scattering detector followed by a refractive index detector.

Different light scattering technologies

An introduction to angular dependence

Key Points• At higher molecular size, the intensity of light molecules scatter will start to vary

with measurement angle and we must account for this phenomenon. This is called'angular dependence'

• Isotropic scatterers are small (Rg < ≈15 nm) and have little or no angulardependence in the intensity of the light they scatter.

• Anisotropic scatterers are larger (Rg > ≈15 nm) and the intensity of the light theyscatter varies with angle.

• To accurately measure molecular weight, we must account for this and this is whatdifferent instruments do in different ways.

• If we account for the angular dependence by measuring it (i.e. MALS), we can usethe data to calculate Rg

DetailsThe Rayleigh equation shown previously includes the term 1/Pθ, which so far we haveassumed we know the values for. However, this is actually not true and it is here thatall of the differences between different light scattering instruments come into play.

Pθ, or form factor, is related to the size of the molecule and the angle at which thescattering is determined. This term is defined as:

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Where:

• n0 is the refractive index of the solvent,• Rg is the molecule's radius of gyration,• λ0 is the laser wavelength in a vacuum,• θ is the measurement angle.

It can be seen from the equation that 1/Pθ is dependent on a number of factors to dowith both the sample and the measurement. These include the solvent refractive index(n0), the laser wavelength in a vacuum (λo), the measurement angle (θ) and the size ofthe molecule we are measuring (Rg).

This means that for larger molecules, the amount of light scattered will depend onthe measurement angle. This is called 'angular dependence' and the reason for thisphenomenon is that as the molecular size increases the scattered photons no longerscatter independently, but constructively and destructively interfere with each other asshown in the figure 2.

Figure 2: A. An isotropic scatterer is small relative to the wavelength of the light and scatters light evenly in alldirections. B. An anisotropic scatterer has significant size compared with the wavelength of the incident light andscatters light in different directions with different intensities.

When the molecule is small with respect to the laser, as in figure 2A, it acts as a pointscatterer. In this situation the light is scattered by the molecule with even intensity in alldirections. These molecules are called 'isotropic scatterers'.

However, as the molecule grows in size with respect to the laser, the molecule'ssize and structure starts to become important. Individual photons from the laser arescattered by different points within the molecule. This is dependent on Rg. Thesescattered photons vary in phase so will interfere with each other and the resulting

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measured intensity will depend on the observation position. These molecules arecalled 'anisotropic scatterers'. At the sizes we are interested in, all of the interferenceis destructive which means that the intensity will always appear to be lower than if weassumed it was an isotropic scatterer. If we were to use this intensity to calculate themolecular weight using the Rayleigh equation, we would therefore underestimate it.

In order to remove this effect, the Rayleigh equation tells us that if we were able tomake a measurement of the intensity of the scattered light at a θ of 0°, then sin2(θ/2)would be 0 and 1/Pθ would be 1. So at 0° the intensity of scattered light would beunaffected by the interference and we could relate the intensity to the molecularweight in the same way we can with smaller molecules, measuring at a 90° angle.Unfortunately, we cannot do this as the intensity of the illuminating laser at 0° is somuch stronger than the scattered light that we cannot separate out the purely scatteredlight. As we cannot measure directly at 0° we need to find an alternative technique.

The rule of thumb to use when deciding where angular dependence starts is tosay that, at a diameter below 1/20th of the laser wavelength, molecules scatterisotropically, and above this level you have anisotropic scattering. Most instrumentsusually have lasers with wavelengths between 633 and 670 nm (red) so the limit ofisotropic scattering is between 15.8 and 16.8 nm radius (Because of the dependenceon laser wavelength, the same calculation for a laser with a wavelength of 532 nm(green) will give a value of 13 nm).

This nominal value of around 15 nm radius for the onset of measurable angulardependence will also be the lower limit at which you can still reliably measure themolecular size, Rg . (See "How do we get the molecular size, Rg, from the data?"below)

The different types of light scattering instrument are designed to overcome the angulardependence issue in different ways and these are described in the next section.

How does angular dependence affect our calculations?

Key Points• We must infer the scattering at 0° based on the scattering at other angles• A Zimm plot of KC/Rθ vs sin2(θ/2) can be used to do this.• The molecular size, Rg, can also be determined from the plot

DetailsIn order to overcome the angular dependence, we must decide what the intensity of thescattered light is at 0° based on data from other angles. The best way to calculate andrepresent this is in a 'Guinnier Plot', which is a simplification of a Zimm plot where thesecond virial coefficient is not required.

A Guinnier plot is a plot of KC/Rθ as a function of angle (sin2(θ/2)). Other plots andmodels are available but this is the most common. An example is shown in the figure 3.

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Figure 3: A Guinnier plot shows KC/Rθ as a function of sin2(θ/2). The intercept of the line is 1/Mw and the initial

slope of the line is related to Rg.

Measurements of light scattering intensity are made at the desired angle. Withknowledge of the sample's concentration and other factors,

KC/Rθ can then be calculated and is plotted in the graph as a function of sin2 (θ/2). They-intercept (which equates to an angle of 0°) is 1/Mw from which the molecular weightcan be calculated easily.

How do we get the molecular size, Rg, from the data?

Key Points• The molecular size, Rg, can be determined from the initial slope of the angular

dependence plot• There must be sufficient angular dependence to see a slope above the noise

DetailsThe initial slope of the line is:

Rg can then be calculated from this equation with relative ease. However, at smallmolecular sizes, the slope will be very small and hidden in the noise in the data. Onlywhen the molecule gets to an appreciable size compared to the wavelength of the laserlight will the slope be large enough to obtain a reliable Rg value. As described above,the nominal threshold for a 633nm laser is an Rg of ≈15 nm. However, the absolutelimit for measuring Rg may be lower or higher than this nominal 15 nm, dependingon the sample, solvent and the conditions of measurement. Many MALS instrumentmanufacturers put a limit of 10 nm (or even lower) on their instrument specifications.

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This is often for standard polymers of high dn/dc values and at high concentrationsrun under ideal conditions. In practice, this limit will not be achievable on all samples.At the other extreme, with samples of poor dn/dc run under non-ideal conditions, thesmallest Rg that can be measured may be much higher than the nominal 15nm value.

Light scattering instrumentsThere are 4 different types of SLS instruments available. RALS, LALS, Hybrid RALS/LALS and MALS.

Right-angle light scattering (RALS)

Key points• Measure only at 90°,• Have the best signal-to-noise ratio and sensitivity,• Have the smallest flow cells,• Can only measure molecular weight accurately for molecules with Rg < ≈15 nm,• Are excellent for measuring protein molecular weight,• Cannot measure Rg.

DetailsA RALS device is the simplest light scattering instrument available. It measuresthe intensity of the light scattered at 90° to the incident beam as in figure 4A. Themolecular weight of the sample is then calculated directly from the intensity measuredand the sample concentration. In the Zimm plot in figure 4B the intensity of thescattered light is measured far from the y-axis and it is assumed that the samplescattering is isotropic (i.e. equal at all angles).

Figure 4: A. Schematic of a RALS detector showing the flow passing through the flow cell. Light from the laserenters at the end of the cell and is collected at 90°. B. When using RALS, the Debye plot is reduced to a singlepoint which is assumed to be equal at every angle and therefore equal 1/Mw.

A RALS system has a number of advantages:

• It is a very straightforward instrument to build without any complicated optics. Allthat is required is a flow cell with a window on the side which allows the cell to havea low volume.

• Since the light is passing through the window/liquid interface at 90°, any flare ornoise created by the change in refractive index is minimised.

• As a consequence, a RALS detector is the simplest form of SLS detector and,thanks to the low noise, it has the best signal-to-noise ratio and the best sensitivity.

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However, RALS detectors have some limitations:

• Since the only measurement is at 90°, the assumption is that the intensity of thescattered light is the same at this angle as it is as 0°. This is clearly not true for anymolecule large enough to display any form of significant angular dependence inthe light that it scatters. Therefore a RALS detector cannot accurately measure themolecular weight of any molecule with an Rg > ≈15nm, as discussed in the earliersection.

• As the measurement is only being made at one angle, the slope of the line on theZimm plot cannot be measured so it is not possible to calculate Rg.

Since proteins are almost always smaller than 15nm radius, and scatter light onlyweakly, they require a sensitive detector. Hence RALS detectors are excellent formeasuring protein molecular weight.

Low-angle light scattering (LALS)

Key points• Measure at the lowest angle possible,• Measure the molecular weight of all molecules with the highest accuracy,• Have low volume flow cells,• Cannot measure Rg.

DetailsA LALS detector measures the intensity of light scattered at an angle that is as closeas possible to 0° as shown in figure 5A. This has the advantage that the intensitywill be very close to the intensity at 0° and so the calculated molecular weight will bevery close to the actual molecular weight of the molecule. To be considered a LALSdetector, the angle must be less than 10° and the most common LALS detectors havea measurement angle of 7° but the closer the measurement to 0°, the more accuratethe measurement will be. This will be true irrespective of the size of the molecule thatwe are measuring. Looking at the Zimm plot (figure 5B), a LALS measurement isvery close to the axis and therefore very close to the y-intercept. This means that theerror by measuring at 7° is very low. With a LALS measurement at 7°, sin2(θ/2) is just0.0037, which equates to less than 1% error in the MW even for the largest molecules.

Figure 5: A. Schematic of a LALS detector showing the flow passing through the flow cell. Light from the laserenters at the end of the cell and is collected through the same exit window at a low angle e.g. 7°. B. When usingLALS, the Debye plot is reduced to a single point which is very close to the y-axis and therefore equal 1/Mw forall molecules.

A LALS system has a number of advantages:

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• LALS measures the intensity of light scattering very close to the axis of the Zimmplot so the calculated molecular weight has the highest accuracy.

• The LALS principle applies to the measurement of the molecular weight of anymolecule.

• A dedicated LALS detector can have a small flow cell since it only has toaccommodate one measurement angle.

LALS detectors also have a number of limitations:

• Since the scattered light is only being measured at a single angle, a LALS systemcannot measure Rg.

• LALS detectors do not quite have the sensitivity of RALS detectors making themless sensitive for weakly scattering samples. This is not usually an issue as LALSdetectors are most often used for larger molecules.

• It is more difficult to build a LALS detector as the scattered light and the laserlight are very close to each other so the scattered light must be measured withoutcollecting the incident laser light as well. In addition, LALS detectors are sensitive tocontamination because contaminating particles are usually large and large particlesscatter predominantly in the forward direction. As a consequence of both of thesefactors, early LALS detectors were very noisy and difficult to work with. Similarly,the lower angles on MALS detectors are often noisy for the same reason. However,modern dedicated LALS instruments are able to isolate the scattered light from theincident light thereby minimising the noise and achieving much better sensitivity.

Overall, LALS measurements offer an accurate way to measure molecular weightusing static light scattering. LALS is particularly useful for measuring the molecularweight of large anisotropic scatterers such as synthetic and natural polymers.

RALS/LALS hybrid detectors

Key points• RALS/LALS hybrids combine RALS and LALS detectors into a single unit,• Use RALS to maximise sensitivity for weak scatterers,• Use LALS to maximise the accuracy for anistropic scatterers,• Maintain the small flow cells of RALS and LALS detectors,• Enable Rg to be estimated.

DetailsBased on the discussion above, it is clear that RALS and LALS detectors arecomplementary RALS detectors have the sensitivity to measure small isotropicscatterers and LALS detectors can measure the molecular weight of large moleculeswith unrivalled accuracy. A RALS/LALS hybrid therefore combines these twomeasurements into a single cell as shown in figure 6A. On the Zimm plot, the RALSdetector is used to measure molecules whose size is below the ≈15 nm thresholdof isotropic scattering. The LALS detector is used to accurately measure moleculeswhose size is above the ≈15 nm threshold of isotropic scattering. Software switchesautomatically from LALS to RALS according to the detector signals. For samples thatscatter anisotropically, the ratio of the two calculated molecular weights can be used toestimate Pθ. This in turn can be used to estimate a value for Rg assuming a structuralmodel such as a random coil or a hard sphere.

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Figure 6: A. Schematic of a RALS/LALS hybrid detector showing the flow passing through the flow cell. Lightfrom the laser enters at the end of the cell and is collected at 90° and through the same exit window at a lowangle e.g. 7°. B. When using RALS/LALS, the Debye plot is reduced to two points where the RALS value isused to maximise sensitivity for isotropic scatterers and the LALS value is used for the most accurate molecularweight for anisotropic scatterers.

The RALS/LALS combination offers the advantages of both:

• The RALS detector offers the best sensitivity for smaller molecules.• The LALS detector offers the best sensitivity for larger molecules.• The two angles can be compared simultaneously in software to get the best data at

every point on the chromatogram.• A RALS/LALS hybrid can maintain the small flow cell associated with both

detectors.• By combining the data from both angles, it is possible to measure the slope of the

line on the Zimm plot and calculate a good estimate of Rg.

Limitations of the RALS/LALS hybrid include:

• LALS data can be noisy if the GPC/SEC system is not clean• The Rg calculation has a limited accuracy.

Overall, a RALS/LALS hybrid offers the advantages of RALS and LALS and suffersnone of the drawbacks of either. This makes a RALS/LALS hybrid system excellent formeasuring the molecular weight of any sample.

Multi-angle light scattering (MALS)

Key points• Measures the intensity of light scattered at multiple angles and extrapolates back to

0°.• Measurements at multiple angles can increase confidence in the data.• Angles can be removed if required.• Measures the molecular weight of molecules of all sizes.• Measures the Rg of molecules > ≈ 15 nm.• It is difficult to know how to select the fit model.• The design of the optics means MALS systems have larger flow cells.• The complexity of the optical design can introduce noise and reduce accuracy• The optical complexity increases the cost

DetailsAs the name suggests, MALS detectors measure the intensity of the scattered lightat many angles as can be seen in figure 7A. By plotting these points on a Zimm plot(figure 7B), a best fit line can be extrapolated back to 0° from where the molecular

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weight can be calculated. The initial slope of this line enables an accurate calculationof molecular size, Rg.

Figure 7: A. Schematic of a MALS detector showing the flow passing through the flow cell. Light from thelaser enters at the end of the cell and scattered light exits at different angles. The scattered light is collectedby multiple detectors. B. When using MALS, the Debye plot is completed and extrapolated back to 0°. Themolecular weight is calculated from the intercept and Rg is calculated from the initial slope of the line.

MALS detectors have a number of their own advantages:

• By measuring the intensity of the scattered light at many angles, the user canbe more confident in the results from any given angle by comparing it to itsneighbouring angles.

• MALS measurements can measure the molecular weight of all molecules fromsmall to large since any angular dependence in the light scattering is alwaysaccounted for.

• By measuring at many angles, it is possible to make an accurate measurement ofRg.

• By studying the shape of the line in the Zimm plot, MALS offers insights into theangular dependence of the scattered light.

• If any angle fails or includes too much noise, it can be removed without seriouslyaffecting the result

However, MALS systems also have some limitations:

• Since the exact shape and structure of the molecule are unknown, it is difficult toknow which extrapolation fit and model will give the correct answer.

• In order to incorporate more angles, MALS flow cells must have a larger volumeresulting in increased peak broadening

• The complexity of the cell means lower angles are often noisier than a RALS orLALS measurement which can reduce the measurement accuracy.

When making measurements using MALS, the most important factor is that the formof the extrapolation is unknown and accurate fitting is particularly dependent on thenumber of low angles and the accuracy of the points. It is therefore important that aMALS instrument should have as many low angles as possible as this will provide themost accurate extrapolation back to 0°. In general, in order to maximise the accuracyof the extrapolation, having more angles is beneficial.

In summary, MALS offers a universal solution for measuring molecular weight and Rgof all different types of sample but the nature of the design forces some compromiseson the measurement.

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Summary of the benefits of MALS, LALS, and RALSsystems• LALS is the best measurement technique for accurate molecular weight

measurements of large (Rg > ≈10-15 nm radius) molecules because it minimisesthe effect of the angular dependence of the scattered light and eliminates the needfor extrapolation.

• Small molecules scatter light isotropically, meaning that there is no angulardependence. In this situation, a single measurement at 90° (RALS) offers the mostsensitive and accurate measurement for these weakly scattering molecules. Thisincludes almost all protein applications.

• MALS measures molecular weight over the whole size range.• MALS offers insights into the angular dependence of the scattered light for large

molecules (>10-15 nm radius) enabling the highest quality and most accuratemeasurement of Rg when compared with a RALS/LALS hybrid. Rg can be used tocharacterise structure for these large molecules.

Calibration

Are static light scattering measurements 'absolute'?Light scattering techniques are often called 'absolute' but this is often misunderstood.

• The addition of a light scattering detector to an SEC system frees us from all formsof column calibration and allows us to measure the molecular weight of the sampledirectly without generating a calibration curve. In this respect we often call thetechnique absolute.

• From a more fundamental perspective, the theory described above comes from firstprinciples with no assumptions and so, again, in this respect it is often describedas an 'absolute' technique. However, even though the method derives from firstprinciples, at some point in the process, we still have to relate the magnitude of thesignal from a photo-detector (i.e. the intensity of the scattered light) to molecularweight of the molecule doing the scattering. This will always require some form ofinstrument calibration or the generation of response factors etc. so the techniquecannot really be said to be truly 'absolute'.

Principle of instrument calibrationAlthough the column doesn't need calibrating, all light scattering devices requirecalibration in some form. The only question is how the device should be calibrated andthis can be done in a number of ways:

The principle of light scattering calibration, as with all calibrations, is to reference thesize of the response signal to the parameter being measured. In static light scattering,we relate the magnitude of the signal from the photo-detector to the intensity of thescattered light and then to the molecular weight of the sample. As well as all thesample dependent factors described in the Rayleigh equation, the measured intensityof the scattered light will depend on a large number of instrument dependent variablesincluding:

• Incident laser power,• Detector sensitivity,

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• Scattering volume (the intersection volume of the laser beam and the detectionoptics),

• The distance from the scattering volume to the detector,• Flare and refraction effects at the interface between the cell and the windows.

It is not possible to characterise all of these components in a system to the accuracyrequired of the measurement; so instead they are all accounted for by a calibrationstep.

There are two methods by which a system can be calibrated but they are functionallyidentical.

Molecular weight standard-based calibration

Key points• Calibration with a molecular weight standard calibrates all light scattering detectors

at the same time.• Calibration in this way is done under the conditions of the measurement.• This method also allows other detectors calibration constants, inter-detector

volumes, and peak broadening corrections to be calculated simultaneously.• This method allows the calibration to be checked and repeated at any time.

DetailsThe principle behind a molecular weight standard-based calibration is to run a sampleof known molecular weight through the SLS detector while it is attached to a GPC/SEC system. The magnitude of the response of the light scattering detector and theknown sample concentration, molecular weight and dn/dc are used to calculate aninstrument response factor or calibration constant for the light scattering detector.Discounting other values such as the solvent refractive index and laser wavelength, thelight scattering detector calibration constant is defined as:

Where:

• kls = the light scattering calibration constant,• δLSstd = measured intensity of light scattered by the standard,• Mwstd = the molecular weight of the standard,• dn/dc = is the standard refractive index increment,• massstd = the injected mass of the standard.

With a standard, the concentration and injection volume (and therefore mass) areknown, as is the standard molecular weight and dn/dc.

This allows the calibration constant to be calculated and can be performed on everyangle in a light scattering detector simultaneously.

Once the calibration constant has been determined, the molecular weight for anysample can be measured:

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Where:

Mw = the molecular weight of the sample,

δLS = measured intensity of light scattered by the sample,

dn/dc = is the sample refractive index increment,

mass = the injected mass of the sample,

kls = the light scattering calibration constant.

In a GPC/SEC system containing multiple detectors, all of them can be calibrated atthe same time using a standard that is well characterised and traceable. A furtherbenefit of calibrating the detectors in this way is that the inter-detector volumesbetween the different detector flow cells and the band broadening and tailing that occuras the sample travels between detectors can all be accounted for in the same step.

The final benefit of this method is that the system calibration can be checked and/or re-calibrated at any time by the user. This is particularly useful as instrument constantswill drift over time due to a number of factors such as degradation of light sources,cleanliness of flow cells, changing detector tubing etc.

A molecular weight standard-based calibration therefore allows all of the instrumentresponse factors/calibration constants to be derived at the same time as the inter-detector volumes and peak broadening parameters are all calculated. It offers the mosteffective way of calibrating a system and, by its very nature, it also offers a clear way tocheck the calibration of a system by comparing the responses of the standard to that atthe time of calibration.

Scattering standard-based calibration

Key points• Calibration with a scattering standard compares the scattering from the sample to

that of a standard with a known Rayleigh ratio.• This calibration only calibrates the 90° detector. Other detectors must be

'normalised' using a second step with another standard.• Inter-detector volumes, and peak broadening and tailing calculations must all be

performed separately

DetailsThe principle behind a scattering standard-based calibration is to fill the light scatteringflow cell with a solvent of known scattering such as toluene. Toluene has a wellcharacterised Rayleigh ratio (the ratio of scattered light intensity to incident lightintensity) so by measuring the scattering signal observed from toluene it is possibleto calculate the scattering signal from the sample. Rθ for the sample is calculatedaccording to the equation:

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Where:

• Rθ = the sample Rayleigh ratio,• IA = the scattered light intensity from sample,• n0 = the refractive index of the sample solvent,• IT = the intensity of light scattered by toluene,• nT = the refractive index of toluene,• RT = the Rayleigh ratio of toluene.

The sample molecular weight is then calculated by inputting this value of Rθ intothe Rayleigh equation. This means that the calibration constant in this situation isessentially:

Although you do not need a molecular weight value to calibrate a detector, the principleof calibrating with a scattering standard is identical in that you are using a standard tomeasure the response of the light scattering detector to a known sample.

This method has a number of other significant drawbacks:

• Since the scattering standard is usually a different solvent to that used to makemeasurements, and therefore has a different refractive index, it can only be usedto calibrate the 90° detector. All of the other detector responses in a systemcontaining more than one angle must subsequently be normalised against the90° detector. This requires running a standard through the system and acquiringnormalisation factors for the non-90° detectors. A normalisation factor is simply amultiplier of the 90° calibration constant, so is functionally just another calibrationconstant.

• Secondly, a scattering standard-based calibration does not allow you to correctfor inter-detector volumes or peak broadening or tailing so this must also be doneseparately, either with the normalisation or as another step.

• Since a scattering standard-based calibration requires taking the light scatteringdetector completely off-line, it is a much more labour intensive process, oftenrequiring the unit to be returned to the manufacturer for calibration. This also meansthat it is very difficult to check that the calibration is still valid.

Overall then, a scattering standard-based calibration is a more labour intensiveprocess that does not improve the calibration and only serves to add more hurdles toachieving an accurate calibration of a system.

Useful references• Kratochvil, P. (1987) Classical Light Scattering From Polymer Solutions. Elsevier• Striegel, A.M. et al. (2009) Modern Size Exclusion Chromatography. John Wiley &

Sons

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